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Stochastic Asset Model
A stochastic investment model tries to forecast how returns and prices on different assets or asset classes, (e. g. equities or bonds) vary over time. Stochastic models are not applied for making point estimation rather interval estimation and they use different stochastic processes. Investment models can be classified into single-asset and multi-asset models. They are often used for actuarial work and financial planning to allow optimization in asset allocation or asset-liability-management (ALM). Single-asset models Interest rate models Interest rate models can be used to price fixed income products. They are usually divided into one-factor models and multi-factor assets. One-factor models * Black–Derman–Toy model * Black–Karasinski model * Cox–Ingersoll–Ross model * Ho–Lee model * Hull–White model * Kalotay–Williams–Fabozzi model * Merton model * Rendleman–Bartter model * Vasicek model Multi-factor models * Chen model * Longstaff–Schwartz model ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Of Return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, cash dividends, stock dividends or the payoff from a derivative or structured product. It may be measured either in absolute terms (e.g., dollars) or as a percentage of the amount invested. The latter is also called the holding period return. A loss instead of a profit is described as a '' negative return'', assuming the amount invested is greater than zero. To compare returns over time periods of different lengths on an equal basis, it is useful to convert each return into a return over a period of time of a standard length. The result of the conversion is called the rate of return. Typically, the period of time is a year, in which case the rate of return is also called the annualized return, and the conversion process, described below ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hull–White Model
In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates onto a tree or lattice and so interest rate derivatives such as bermudan swaptions can be valued in the model. The first Hull–White model was described by John C. Hull and Alan White in 1990. The model is still popular in the market today. The model One-factor model The model is a short-rate model. In general, it has the following dynamics: :dr(t) = \left theta(t) - \alpha(t) r(t)\right,dt + \sigma(t)\, dW(t). There is a degree of ambiguity among practitioners about exactly which parameters in the model are time-dependent or what name to apply to the model in each case. The most commonly accepted naming convention is the following: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilkie Investment Model
The Wilkie investment model, often just called Wilkie model, is a stochastic asset model developed by A. D. Wilkie that describes the behavior of various economics factors as stochastic time series. These time series are generated by autoregressive models. The main factor of the model which influences all asset prices is the consumer price index. The model is mainly in use for actuarial work and asset liability management. Because of the stochastic properties of that model it is mainly combined with Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...s. Wilkie first proposed the model in 1986, in a paper published in the ''Transactions of the Faculty of Actuaries''. It has since been the subject of extensive study and debate. Wilkie himself updated and expanded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Brownian Motion
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model. Technical definition: the SDE A stochastic process ''S''''t'' is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): : dS_t = \mu S_t\,dt + \sigma S_t\,dW_t where W_t is a Wiener process or Brownian motion, and \mu ('the percentage drift') and \sigma ('the percentage volatility') are constants. The former is used to model deterministic trends, while the latter term is often used to model a set of unpredictable events occurring during this motion. Solving the SDE For an arbitrary initial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black–Scholes Model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a ''unique'' price given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited. The main principle behind the model is to hedge the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Model
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resulting d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LIBOR Market Model
The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to the names of some of the inventors) is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic derivatives like Bermudan swaptions, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps and spread options, among many others. The quantities that are modeled, rather than the short rate or instantaneous forward rates (like in the Heath–Jarrow–Morton framework) are a set of forward rates (also called forward LIBORs), which have the advantage of being directly observable in the market, and whose volatilities are naturally linked to traded contracts. Each forward rate is modeled by a lognormal process under its forward measure, i.e. a Black model leading to a Black formula for interest rate caps. This formula is the market standard to quote cap prices in terms of implied volatilities, he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Longstaff–Schwartz Model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate. The short rate, r_t \,, then, is the ( continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t. Specifying the current short rate does not specify the entire yield curve. However, no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of r_t \, as a stochastic process under a risk-neutral measure Q, then the price at time t of a zero-coupon bond maturing at time T with a payoff of 1 is given by : P(t,T) = \operatorname^Q\left \mathcal_t \right where \mathcal is the natural filtration for the process. The inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chen Model
In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" (short-rate model) as it describes interest rate movements as driven by three sources of market risk. It was the first stochastic mean and stochastic volatility model and it was published in 1994 by Lin Chen, economist, theoretical physicist and former lecturer/professor at Beijing Institute of Technology, Yonsei University of Korea, and Nanyang Tech University of Singapore. The dynamics of the instantaneous interest rate are specified by the stochastic differential equations: : dr_t = \kappa(\theta_t-r_t)\,dt + \sqrt\,\sqrt\, dW_1, : d \theta_t = \nu(\zeta-\theta_t)\,dt + \alpha\,\sqrt\, dW_2, : d \sigma_t = \mu(\beta-\sigma_t)\,dt + \eta\,\sqrt\, dW_3. In an authoritative review of modern finance (''Continuous-Time Methods in Finance: A Review and an Assessment''), the Chen model is listed along with the models of Robert C. Merton, Oldrich Vasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vasicek Model
In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives, and has also been adapted for credit markets. It was introduced in 1977 by Oldřich Vašíček, and can be also seen as a stochastic investment model. Details The model specifies that the instantaneous interest rate follows the stochastic differential equation: :dr_t= a(b-r_t)\, dt + \sigma \, dW_t where ''Wt'' is a Wiener process under the risk neutral framework modelling the random market risk factor, in that it models the continuous inflow of randomness into the system. The standard deviation parameter, \sigma, determines the volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters b, a and \sigma, tog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rendleman–Bartter Model
The Rendleman–Bartter model (Richard J. Rendleman, Jr. and Brit J. Bartter) in finance is a short-rate model describing the evolution of interest rates. It is a "one factor model" as it describes interest rate movements as driven by only one source of market risk. It can be used in the valuation of interest rate derivatives. It is a stochastic asset model. The model specifies that the instantaneous interest rate follows a geometric Brownian motion: :dr_t = \theta r_t\,dt + \sigma r_t\,dW_t where ''Wt'' is a Wiener process modelling the random market risk factor. The drift parameter, \theta, represents a constant expected instantaneous rate of change in the interest rate, while the standard deviation parameter, \sigma, determines the volatility of the interest rate. This is one of the early models of the short-term interest rates, using the same stochastic process as the one already used to describe the dynamics of the underlying price in stock options. Its main disadvanta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Merton Model
The Merton model, developed by Robert C. Merton in 1974, is a widely used credit risk model. Analysts and investors utilize the Merton model to understand how capable a company is at meeting financial obligations, servicing its debt, and weighing the general possibility that it will go into credit default. Under this model, the value of stock equity is modeled as a call option on the value of the whole company – i.e. including the liabilities – struck at the nominal value of the liabilities; and the equity market value thus depends on the volatility of the market value of the company assets. The idea applied is that, in general, equity may be viewed as a call option on the firm: since the principle of limited liability protects equity investors, shareholders would choose not to repay the firm's debt where the value of the firm is less than the value of the outstanding debt; where firm value is greater than debt value, the shareholders would choose to repay – i.e. exer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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